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Consider a regression study involving a dependent variable $y,$ a quantitative independenvariable $x_{1},$ and a categorical independent variable with three possible levels (level $1,$ level 2and level 3$)$$\begin{array}{l}{\text { a. How many dummy variables are required to represent the categorical variable? }} \\ {\text { b. Write a multiple regression equation relating } x_{1} \text { and the categorical variable to } y. } \\ {\text { c. Interpret the parameters in your regression equation. }}\end{array}$

a) see tableb) $E(y)=\beta_{0}+\beta_{1} x_{1}+\beta_{2} x_{2}$c) $\beta_{0}$$\beta_{0}+\beta_{1}$$\beta_{0}+\beta_{2}$

Intro Stats / AP Statistics

Chapter 13

Multiple Regression

Descriptive Statistics

Linear Regression and Correlation

Missouri State University

Piedmont College

Cairn University

University of St. Thomas

Lectures

0:00

01:02

Consider a regression stud…

01:10

this question. We're told that we have a regression study with why, as the dependent variable, we have a quantitative, independent variable X one and a categorical variable with three possible levels level 12 and three, 12 and three, and were asked in part a how Maney dummy variables are required to represent the categorical variables. So we have three levels in the categorical variables for the number of dummy variables that are needed is always K minus one. So that's three minus one, which gives us two, which is our answer to part A. So in part, we were asked to write a multiple regression equation relating X one and the categorical variable to watch. So relating X one a zwelling as the categorical here. So we know that from part A. We need two dummy variables to represent the categorical variables. So we need two dummy variables x two and x three, which we can basically say represent level 12 and 30001 and 10 So our multiple regression equation here is why it's supposed to be the zero, which is the intercept was beta one x one, which is our quantitative variable plus beta two x two plus data three x three That's our multiple regression equation for this question and the answer to part people I know in part to see whereas to interpret the parameters in our equation. So, actually, let's just change this around. So we have 10 and 01 So in purchasing, you're interpreting the parameters. Yeah, so you had. First, let's look at the different cases. So horror or level one, we have x 20 and x 30 So for level one, we have wise he close to theater zero plus beta one x one for level two. We have wise close to beta zero plus para one x one plus data to next to then for level three. We have wise equals two pita, zero para one x one plus beta three x three So our first interpretation data one. It's basically represents the change in why for a one unit change in X one. If we keep x two and x three construct four Level one for level two. Better X two is actually one, so we basically have paid a zero plus better to plus para one fixed one. So again it's Ah Peter one again. It's the same one unit change in X one keeping X to the next three constant Beta two is just an increase or decreased you to level to. So it's basically the same as expectation off. Why? Given the level two minus the expectation of why given level one? Because that's basically the difference between the first equation and the second equation. And now for the third, we basically have X three as one. So we can simplify this to say Pierre a zero plus beta three plus Peter one x one. So, Peter three is basically the change you to l three and the changes if we look at If you compare it to al one, it's basically why given out three minus affectation of why given album. So it's actually the difference from l one. So there are three is the difference due to l three from l one invaded to It's the difference due to help to from one and beta one is just the amount of changing. Why, for a unit change in X, which is our answer to hard see

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